@article{oshaughnessy.22b,
    author = 	 {O'Shaughnessy, M. and Davenport, M. and Rozell, C. },
    title = 	 {Distance preservation in state-space methods for detecting causal interactions in dynamical systems},
    year =	 2023,
	month = aug,
	abstract = {The popular ``state-space'' class of algorithms for detecting casual interaction in coupled dynamical systems are inspired by Takens' embedding theorem, which provides conditions under which relationships involving attractors and their delay embeddings are diffeomorphic. In practice, however, state-space methods often test how these relationships preserve distances --- a stronger property than the diffeomorphisms guaranteed by Takens' theorem. In this paper we explore these algorithms explicitly from the perspective of distance preservation, providing both theoretical guarantees applicable to simple systems and empirical demonstrations that caution against applying distance-preservation-based arguments to more complex systems. Our theoretical guarantees, which use recently developed geometry-preserving extensions of Takens' theorem, provide conditions under which the stability of a certain map reveals underlying causal structure. Our empirical results show that typical coupled systems do not satisfy stability assumptions, underlining the importance of other system properties when analyzing the performance of this class of algorithms.},
	url = {https://arxiv.org/abs/2308.06855},
	note = {Under review.}
}